(this readily generalizes to the inhomogeneous case: The factor \(\left|\gamma \right|^{2{^{\sharp }}\hspace{-1.99168pt}\mathcal {F}}\) 
just needs to be replaced by \(\prod _{k\in \mathcal {F}}\left|\gamma _{k}\right|^{2}\) ).
This result makes no use of the fact that the environment components
were spins, and thus it is directly applicable to (pure state) photon
scattering off an object in a two dimensional superposition, more
directly supporting the connection discussed in Touil et al. [1]}
and extending it to \(\chi \left(\mathcal {S}:\check{\mathcal {F}}\right)\)  in the good
decoherence limit Moreover, as with the QCB result, the form of the accessible information
for pure \(\mathcal {S}\mathcal {E}\)  states, \(H_{\mathcal {S}}-h\left(P_{e}\right)\)  , with the
optimal \(P_{e}\)  from Eq. (REF ) holds regardless of the
environment components. They can be spins, qudits, or photons. Furthermore,
the connection with hypothesis testing allows for even more general
statements about models that are not purely decohering. For instance,
for projection-valued measurements and pure \(\mathcal {S}\mathcal {E}\)  states, one obtains
the same accessible information, \(H_{\mathcal {S}}-h\left(P_{e}\right)\) , but
the error probability just has the overlap between the conditional
fragment states, which can behave in a manner that is not exponentially
decaying with \({^{\sharp }}\hspace{-1.99168pt}\mathcal {F}\) ..
